Ladder operators in angular momentum
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- เผยแพร่เมื่อ 5 ก.ย. 2024
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📚 Ladder operators allow us to increase or decrease a component of angular momentum by a discrete amount hbar. These operators are extremely useful because angular momentum is indeed quantized. In this video, we derive a long list of properties of ladder operators that will prove extremely useful in our study of angular momentum.
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⏮️ BACKGROUND
Hermitian operators: • Hermitian operators in...
Compatible observables: • Compatible observables...
Angular momentum: • Angular momentum in qu...
⏭️ WHAT NEXT?
Angular momentum eigenvalues: • Angular momentum eigen...
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Director and writer: BM
Producer and designer: MC
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HELLO, PHYSICS IS BEAUTIFUL WHEN THERE ARE BRILLIANT PEOPLE LIKE YOU.
Thanks!
YOU ARE TOO DUMB TO JUDGE WHETHER SOMEONE IS BRILLIANT, MATE.
BECAUSE SHE DOES NOT RECOGNIZE THIS. SHE HAS HER LIMITS TOO.
I was stuck in this topic for more than a week, and i have tried learning it through different books and lecture notes. In books, the notations were really confusing and some steps were skipped.
I had to pause video many times, it really helped and cleared a lot of confusion. There is no confusion for now atleast, and i feel so relaxed right now.
God bless you professor, love from pakistan.
This is great, glad we could help!
Thank you Professor M. Into Q.M. about two years now. Am getting bitten bad, very interesting. Your teaching is riveting & gain my attention. Good stuff. Daniel Blatecky USA
Glad you like it, and good luck with your learning!
This is the only video in internet, that made me understand ladder operators thoroughly.
Thank you ma'am
This is great to hear! :)
This was more clear than a chapter in Sakurai. Well done
Thanks for your kind comment, and glad you enjoy it! :)
your channel is criminally underrated, hope you go viral soon
Thanks for your support!
I was always scared of this topic but you made it crystal clear to me thankyou professor
Glad you found the explanation useful! :)
This channel is GOLD
Glad you like it! :)
If i pass QM paper,that will be only because of this channel ❤
Glad to hear, and good luck with your exam!
i really appreciate , teaching full quantum mech with so much detailed explaination is too much sufficient and interesting for any student to generate a spark to study more of it. :)
thank you so much and i request u to please keep making more and lot like this for different topics too, i am lovin it
Thanks for watching, and glad you like the videos!
Thanks mam....🙏💜,from India...
Glad you like it!
As always, excellent! In particular, I thought the derivation of the normalization constants for the kets produced by action of the ladder operators was very helpful! Thank you!
Great you liked the derivation! :)
1:39 my 1st real QM teacher was Stan "the man" Mandelstam, and he had a great South African accents and would say: Jay eye Jay jay minus Jay jay Jay eye is eye epsilon eye jay kay Jay kay...so often it is as burned into my mind as much as ay squared plus bee squared is sea squared.
He also would say "Dee Toop Sigh Dee feet or phi" for start of the angular momentum operator in position rep.
Great! You guys deserve more subscribers!!!
Thanks for your support! :)
Thank you so much, you saved my course, your videos are so clear, kind regards from Mexico 🙏
Great to hear this! What university are you at?
Ladder operators are not just discrete, they are like ladders because each step is the same size, and you can fall off either end. I know math ppl call that last one nilpotent, but idk where to go with that.
This explanation is clear for the most part, but I believe the primary reason behind using ladder operators want quite clear. I still don't understand why they are needed.
Perhaps there could be an example of showing the importance of the concept of ladder operators with working with angular momentum before introducing them.
For example, the QHO has ladder operators for the hamiltonian as we know that there are energy levels and so would make calculation of energies easier if we expressed the operators as a product of two operators where one raises the energy and the other lowers.
Apart from that this was a great video.
Thanks for the feedback! We always try to balance the amount of content with the length of the video, and to keep them in the range 10-25 min we have to present the topics in a somewhat piecemeal fashion. However, the idea is that each video is part of a larger whole, with relevant links in the video description and also in the corresponding playlists. In this particular example of angular momentum ladder operators, we use them thoroughly in the related video on angular momentum eigenvalues, where you can start to see their importance:
th-cam.com/video/t7x6vt6xkAI/w-d-xo.html
They are used for exactly the same reasons as in the harmonic oscillator, which is to factorize the operator you are trying to find the eigenvalues for. In the harmonic oscillator, one factorization works of the form A*A+E. For angular momentum, there are three terms J^2=(J+J-+J-J+)/2+J3^2. Whenever an operator can be factorized in terms of one operator and its adjoint, one can find the eigenvalues purely from algebra. This is very powerful and is why they are employed here. It is just more complicated here because the factorization is into the sum of three factorizations rather than just one. But it can still be done.
This is such excellent content that you have here. I appreciate it. I am not taking any courses in quantum mechanics, but it is something that interests me. I very much like the explanations using Dirac notation to derive algebraically the results that you do. You explain very well not only what you are doing but why.
This is great to hear, thanks! :)
Excellent
Glad you like it!
very helpful video for me. thank you for uploading
Glad you find it helpful!
You are the best!! thanks a lottttt
Glad you like this!
Really well explained! Thanks!!
Glad you like it! :)
Thanks for the clear math ❤
Glad you find it clear!
We kindly suggest to you to address the concept of Clebsch Gordan Coefficients sometime soon. Thanks !
We are planning a full series on angular momentum, including addition of angular momenta, so we should hopefully get to Clebsch Gordan coefficients soon!
@@ProfessorMdoesScience This will be AMAZING ! You guys are turning in reference in QM.
Thanks!
you are the top G!
Thank you so much!!, nice work you got here, I appreciate it 🙂
Glad you like it, and thanks for watching!
Crystal clear! Thanks!
Glad it was clear! :)
Awesome material. Thank you!
Thanks!
Beautifully explained 🙂
Glad you like it! :)
Really good video again - just one question: Why is there a vector accent on J^2? It is not a vector is it?
You are correct that J^2 is not a vector, it is the scalar product of a vector with itself, which overall gives a scalar. We could write in a variety of ways (e.g. J (dot) J, J1^2+J2^2+J3^2, etc), and we use the shorthand notation J^2 as it is one of the most commonly used ones. I hope this helps!
Also do you have a video proving the two red equations at 20:44
Yes, for that what we need is to show that mu=m*hbar and that lambda=j(j+1)*hbar^2. These two results are demonstrated in the video on eigenvalues:
th-cam.com/video/t7x6vt6xkAI/w-d-xo.html
I hope this helps!
great video, thanks
Thanks for your continued support! :)
Thanks a lot for all these great videos. Do you also have a video where you explain why mu is m times hbar and lambda = j(j+1)hbar^2?
We do! You can find the details in this video: th-cam.com/video/t7x6vt6xkAI/w-d-xo.html
In general, if you follow the videos in the order in which they appear in the various playlists, then we build the knowledge step by step:
th-cam.com/users/ProfessorMdoesScienceplaylists
@@ProfessorMdoesScience ah that was a struggle for me too. i think you have ladder operators before the angular momentum eigenvalues video in the playlist. At least that's the order it shows up to me.
Smashed the like button!
Glad you like it! :)
Hi there, i understand that we need a normalization constant N₊ so that we can set J₊|λ,µ> = N₊ |λ,µ+ħ> (15:55). But why exactly is that given by the norm of J₊|λ,µ> ? (18:30)
Thank you for the great video!
I also don't see how J₊J_ =J²-(J_3)²+ħJ_3 is derived
For the derivation of J+J-, the first step is:
J+J- = (J1+iJ2)(J1-iJ2)
This simply uses the definitions of J+ and J- in terms of J1 and J2. The next step is:
(J1+iJ2)(J1-iJ2) = J1^2 + J2^2 + i(J2J1-J1J2)
This is obtained by multiplying through the two terms in the left hand side, just like you would (a+b)(a-b) = a^2 + b^2 + ba - ab, and note that J2J1 is not equal to J1J2 as the operators don't commute. The next step is realising that the last term in the expression above is simply the commutator of J1 and J2:
i(J2J1-J1J2) = -i(J1J2-J2J1) = -i[J1,J2]
so that we end up with:
(J1+iJ2)(J1-iJ2) = J1^2 + J2^2 - i[J1,J2]
The next step is to realise that [J1,J2] = i*hbar*J3, a result we derive in this video:
th-cam.com/video/Bo5qoaLsBOE/w-d-xo.html
This means we end up with:
(J1+iJ2)(J1-iJ2) = J1^2 + J2^2 - i*(i*hbar*J3)
Using i^2=-1, we have that -i*(i*hbar*J3) = hbar*J3
This means that:
(J1+iJ2)(J1-iJ2) = J1^2 + J2^2 + hbar*J3
Finally, by definition, J^2 = J1^2 + J2^2 + J3^2, so that we can write J1^2 + J2^2 = J^2 - J3^2. Replacing this in the equation above, we get:
(J1+iJ2)(J1-iJ2) = J^2 - J3^2 + hbar*J3
And this is the end of the derivation.
I hope this is clearer, and in general I would recommend watching the videos in the order provided by the corresponding playlists, so that you are familiar with any results we quote.
For the norm question, the key is that the eigenstates |lambda,mu> and |lamba,mu+hbar> are normalised, so they have norm one. Therefore, if we apply the operator J+ on |lambda,mu>, we may in general end up with a state with a different norm, and we call that state N+|lambda,mu+hbar>. This means that |N+|^2 gives the norm of the new state. I hope this helps!
@@ProfessorMdoesScience Thank you for the detailed answer! i watched the other videos, but this identiy in particular was trickier to me. The second one is also clear now!
@@khedot6291 Glad to hear it is clearer now! :)
Thank you
Thanks for watching!
Which video that you mentioned in the video has the eigenvalue of the angular momentum operators?
You can find the link in the video description, but it is this one: th-cam.com/video/t7x6vt6xkAI/w-d-xo.html
15:58 I'm having a hard time to understand the ket notation | lambda, mu+h>, how come ? Is it the same reason as for the translation operator ? Could you please detail me a similar exemple, I would appreciate it ! Thank you in advance ! Outstanding vid btw
Glad you like it! In principle we could label kets in any way we find convenient. When we have eigenkets, we typically label them using the associated eigenvalue. For example, if we have an eigenvalue "a" for an operator A, then we write the associated eigenket as |a>, and the eigenvalue equation would be A|a>=a|a>. In the video, the states are simultaneous eigenkets of two operators (J^2 and Jz), so we use the two corresponding eigenvalues for the labels, separated by a comma. Specifically, |lambda,mu+h> means that this is an eigenket with J^2 eigenvalue lambda and with Jz eigenvalue mu+h. I hope this helps!
@@ProfessorMdoesScience Wonderful explanation ✨! Thanks you a lot!
At 17:35 in the video you introduce an equation for the norm. How is this derived?
The norm of a vector (ket) is related to its length, calculated using the scalar product of the vector with itself. We introduce the basics behind these ideas in this video:
th-cam.com/video/hJoWM9jf0gU/w-d-xo.html
I hope this helps!
In the playlist "angular momentum", this video and the next (the one on eigenvalues) seem to be reversed.
We actually have them in this order on purpose. The idea is that we first introduce the ladder operators, and we show that, in general, they raise/lower the mu eigenvalue by a factor of hbar, where mu is kept general. The derivation of this result does not require that mu has the actual value it takes for angular momentum, hence we can do this derivation first. With this result, we can then figure out the actual values that lambda and mu take in the next video (the one on eigenvalues). I hope this helps!
@@ProfessorMdoesScience I see! What got me confused is that at 11:39 you say "As we discuss in the video about eigenvalues, it turns out..." and I misheard it as "as we descussED" (and I think there was some more places where you mention results that are derived in the other video, so at some point I was under the impression I was supposed to have already watched that!). Thank you!
BTW these videos are AMAZING!
Pls try and use dirac notation
We do actually use Dirac notation often, see for example the first video where we introduce it:
th-cam.com/video/hJoWM9jf0gU/w-d-xo.html
I hope this helps!
You guys should get more subscribeers ❤️
Thanks for watching! We are growing steadily, and one way you can support us further is by telling your friends! :)
@@ProfessorMdoesScience alright I'll ❤️
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Do a series teaching me stuff! Just live stream lessons. My wife had a religious education and I had a crappy us public education. My dad was a pharmaceutical biochemist, i would thrive with a good instructor.
We are hoping to expand the resources we provide, perhaps including live sessions, but it will still take a while for us to get there. In the meantime, I hope you enjoy the videos!
@@ProfessorMdoesScience you're amazing, thank you for the outstanding content!
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AGAIN, THIS IS NOT AN OPERATOR: 1:06. IT IS A VECTOR AND PLAYS NO ROLE IN QM. ONLY ITS COMPONENTS ARE OPERATORS (AS WELL AS L^2) !!!
you are so beautiful